What is the probability that all three selected from four men and six women will be of the same gender?

To determine the probability that all three selected individuals are of the same gender from a group of four men and six women, we begin by calculating the total number of ways to select three individuals from the entire group of ten people (four men + six women).

The total number of combinations to choose three people from ten is found using the combination formula:

[

\binom{n}{k} = \frac{n!}{k!(n-k)!}

]

Thus, the total combinations for selecting three from ten is:

[

\binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120

]

Next, we analyze the successful outcomes where all selected individuals are of the same gender. This can happen in two scenarios: selecting all men or selecting all women.

1. Selecting all men: The number of ways to select three men from four is:

[

\binom{4}{3} = 4

]

2. Selecting all women: The number of ways to select three women from six is:

[